[Paper Review] Logarithmic Kodaira-Akizuki-Nakano vanishing and Arakelov-Parshin boundedness for singular varieties

Sándor J. Kovács|arXiv (Cornell University)|Mar 3, 2000

Algebraic Geometry and Number Theory22 references3 citations

TL;DR

This paper establishes a logarithmic Kodaira-Akizuki-Nakano vanishing theorem for singular varieties, generalizing classical results to singular settings with semi-ample and big line bundles. The key contribution is a foundational vanishing result that enables a boundedness result of Arakelov-Parshin type for families of varieties over curves, proving that the degree of the pushforward of pluricanonical bundles is bounded in terms of genus and singular fiber count.

ABSTRACT

The article has two parts. The first part is devoted to proving a singular version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault and Viehweg. This is then used to prove other vanishing theorems. In the second part these vanishing theorems are used to prove an Arakelov-Parshin type boundedness result for families of canonically polarized varieties with rational Gorenstein singularities.

Motivation & Objective

• To generalize the logarithmic Kodaira-Akizuki-Nakano vanishing theorem to singular varieties, extending results from smooth settings.
• To establish a boundedness result of Arakelov-Parshin type for families of varieties over curves with controlled singular fibers.
• To prove that the degree of the pushforward of pluricanonical bundles is bounded in terms of the genus and number of singular fibers.
• To provide a vanishing theorem that serves as a key technical tool for moduli space boundedness and automorphism group estimates.
• To demonstrate that the vanishing theorem implies boundedness even when the obvious generalization fails.

Proposed method

• Introduces a logarithmic vanishing theorem for singular varieties using a relative version of the Kodaira-Akizuki-Nakano vanishing, adapted to singular settings.
• Applies the theorem to families of varieties over a curve $ C $, with $ f: X \to C $, and uses the relative base point free theorem to reduce to a known case.
• Uses the nef and big condition on the relative canonical bundle $ \omega_{X/C} $ to apply vanishing theorems via pullbacks and modifications.
• Constructs a finite cover $ \tilde{X} \to X $ and a modified line bundle $ \tilde{\mathcal{L}} $ that is semi-ample and ample with respect to the complement of a divisor, enabling the application of vanishing.
• Employs the degree of the pushforward $ \deg(f_* \omega_{X/C}^m) $ as a key invariant, bounded via the genus $ g $ and number of singular fibers $ \delta $.
• Uses the moduli functor $ \mathfrak{D}_h^{(m)} $ and its projective compactification $ \bar{D}_h^{(m)} $ to parametrize families and prove finiteness of such morphisms.

Experimental results

Research questions

• RQ1 Can the logarithmic Kodaira-Akizuki-Nakano vanishing theorem be extended to singular varieties, particularly when the line bundle is semi-ample and big?
• RQ2 Does the boundedness of the degree of the pushforward of pluricanonical bundles hold for families of varieties over curves with $ 2g - 2 + \delta > 0 $?
• RQ3 Is there a finite-type subscheme parametrizing morphisms from a curve to a moduli space of canonically polarized varieties with controlled singular fibers?
• RQ4 Can the vanishing theorem be used to prove boundedness results in the absence of a smooth resolution, particularly in the non-isotrivial case?
• RQ5 What is the sharp bound on the degree of $ f_* \omega_{X/C}^m $ in terms of $ g $, $ \delta $, and $ m $, under the assumption of $ h $-nef and $ h $-big canonical bundle?

Key findings

• The logarithmic Kodaira-Akizuki-Nakano vanishing theorem holds for singular varieties with semi-ample and big line bundles, generalizing Esnault-Viehweg's result to singular settings.
• The degree of the pushforward $ \deg(f_* \omega_{X/C}^m) $ is bounded above by $ 4 \cdot \dim X \cdot (2g - 2 + \delta) \cdot m \cdot e(m) $ for $ m \geq \binom{\dim X}{2} + 2 $.
• The condition $ 2g - 2 + \delta > 0 $ is necessary for boundedness; if $ g = 0 $ and $ \delta \leq 2 $, or $ g = 1 $ and $ \delta = 0 $, boundedness fails due to existence of finite covers that increase degree.
• A finite-type subscheme $ T \subset \mathbb{H} $ parametrizes all morphisms $ \Psi: C \to \bar{D}_h^{(m)} $ such that the family admits a simultaneous resolution over $ C \setminus \Delta $.
• The boundedness of $ e(\omega_F^m) $ on the moduli space $ D_h^{(m)} $ ensures that the degree of the determinant line bundle is bounded, supporting the existence of a coarse moduli scheme.
• The relative base point free theorem allows reduction to a case where the vanishing theorem applies, enabling the proof of boundedness via pullbacks and modifications.